Estimates on Escape Times for the Elephant Random Walk

TL;DR

This paper derives exponential bounds for the escape times of the Elephant Random Walk from symmetric intervals and shows that, in the diffusive regime, its average escape time behaves similarly to a standard symmetric random walk, growing quadratically with the interval size.

Contribution

It provides the first tight exponential bounds for the tail of the escape time of the Elephant Random Walk and demonstrates its diffusive behavior matches classical random walks.

Findings

Exponential bounds for escape time tails are established.

Expected escape time grows quadratically with interval size in the diffusive regime.

Elephant Random Walk exhibits diffusive behavior similar to symmetric random walks.

Abstract

We study the gambler's ruin problem for the Elephant Random Walk, focusing on escape time from a symmetric interval of the form $\{-N, \ldots, N\}$. As our main result, we derive tight exponential bounds for the tail of this escape time. We then illustrate the usefulness of such bounds by proving that, in the diffusive regime, the Elephant's average behavior mirrors that of the traditional symmetric random walk: the expected escape time grows quadratically with $N$.